Optimal. Leaf size=51 \[ \frac{2 a^2 (d x)^{5/2}}{5 d}+\frac{4 a b (d x)^{9/2}}{9 d^3}+\frac{2 b^2 (d x)^{13/2}}{13 d^5} \]
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Rubi [A] time = 0.0142823, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {14} \[ \frac{2 a^2 (d x)^{5/2}}{5 d}+\frac{4 a b (d x)^{9/2}}{9 d^3}+\frac{2 b^2 (d x)^{13/2}}{13 d^5} \]
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin{align*} \int (d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx &=\int \left (a^2 (d x)^{3/2}+\frac{2 a b (d x)^{7/2}}{d^2}+\frac{b^2 (d x)^{11/2}}{d^4}\right ) \, dx\\ &=\frac{2 a^2 (d x)^{5/2}}{5 d}+\frac{4 a b (d x)^{9/2}}{9 d^3}+\frac{2 b^2 (d x)^{13/2}}{13 d^5}\\ \end{align*}
Mathematica [A] time = 0.0111555, size = 33, normalized size = 0.65 \[ \frac{2}{585} x (d x)^{3/2} \left (117 a^2+130 a b x^2+45 b^2 x^4\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 30, normalized size = 0.6 \begin{align*}{\frac{2\,x \left ( 45\,{b}^{2}{x}^{4}+130\,ab{x}^{2}+117\,{a}^{2} \right ) }{585} \left ( dx \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.954147, size = 55, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (45 \, \left (d x\right )^{\frac{13}{2}} b^{2} + 130 \, \left (d x\right )^{\frac{9}{2}} a b d^{2} + 117 \, \left (d x\right )^{\frac{5}{2}} a^{2} d^{4}\right )}}{585 \, d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.21244, size = 86, normalized size = 1.69 \begin{align*} \frac{2}{585} \,{\left (45 \, b^{2} d x^{6} + 130 \, a b d x^{4} + 117 \, a^{2} d x^{2}\right )} \sqrt{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.1575, size = 49, normalized size = 0.96 \begin{align*} \frac{2 a^{2} d^{\frac{3}{2}} x^{\frac{5}{2}}}{5} + \frac{4 a b d^{\frac{3}{2}} x^{\frac{9}{2}}}{9} + \frac{2 b^{2} d^{\frac{3}{2}} x^{\frac{13}{2}}}{13} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12954, size = 57, normalized size = 1.12 \begin{align*} \frac{2}{13} \, \sqrt{d x} b^{2} d x^{6} + \frac{4}{9} \, \sqrt{d x} a b d x^{4} + \frac{2}{5} \, \sqrt{d x} a^{2} d x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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